Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Karen placed 56 bottles into boxes that each held 6 bottles. How many boxes did she use? 10 4 -1
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 (For help, go to Skills Handbook p. 760. ) Find each quotient. 1. 480 ÷ 3 4. 288 ÷ 6 2. 365 ÷ 5 5. 3. 459 ÷ 9 354 2 6. 354 3 Check Skills You’ll Need 4 -1
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Solutions 1. 160 3 480 – 3 18 – 18 0 2. 73 5 365 – 35 15 – 15 0 4 -1 3. 51 9 459 – 45 9 – 9 0
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Solutions (continued) 4. 48 6 288 – 24 48 – 48 0 5. 177 2 354 – 2 15 – 14 0 4 -1 6. 118 3 354 – 3 5 – 3 24 – 24 0
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Is the first number divisible by the second? a. 1, 028 by 2 Yes; 1, 028 ends in 8. b. 572 by 5 No; 572 doesn’t end in 0 or 5. c. 275 by 10 No; 275 doesn’t end in 0. Quick Check 4 -1
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Is the first number divisible by the second? a. 1, 028 by 3 No; 1 + 0 + 2 + 8 = 11; 11 is not divisible by 3. b. 522 by 9 Yes; 5 + 2 = 9; 9 is divisible by 9. Quick Check 4 -1
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 Ms. Washington’s class is having a class photo taken. Each row must have the same number of students. There are 35 students in the class. How can Ms. Washington arrange the students in rows if there must be at least 5 students, but no more than 10 students, in each row? 1 • 35, 5 • 7 Find pairs of factors of 35. There can be 5 rows of 7 students, or 7 rows of 5 students. Quick Check 4 -1
Divisibility and Factors PRE-ALGEBRA LESSON 4 -1 State whether each number is divisible by 2, 3, 5, 9, or 10. 1. 18 2. 90 3. 81 4. 25 2, 3, 9 2, 3, 5, 9, 10 3, 9 5 5. List the positive factors of 36. 1, 2, 3, 4, 6, 9, 12, 18, 36 4 -1
Exponents PRE-ALGEBRA LESSON 4 -2 Which word best completes the statement: sometimes, always, or never? If the product of two factors is zero, both factors are zero. sometimes 4 -2
Exponents PRE-ALGEBRA LESSON 4 -2 (For help, go to Lesson 1 -9. ) Find each product. 1. 3 • 3 • 3 3. (– 4)(– 4) 2. – 12 • (– 12) 4. 10 • 10 Check Skills You’ll Need 4 -2
Exponents PRE-ALGEBRA LESSON 4 -2 Solutions 1. 3 • 3 • 3 = 81 3. (– 4)(– 4) = – 64 2. – 12 • (– 12) = 144 4. 10 • 10 = 10, 000 4 -2
Exponents PRE-ALGEBRA LESSON 4 -2 Write using exponents. a. (– 11)(– 11)4 Include the negative sign within parentheses. b. – 5 • x • y • x – 5 • x • x • y Rewrite the expression using the Commutative and Associative Properties. – 5 x 3 y 2 Write x • x and y • y using exponents. Quick Check 4 -2
Exponents PRE-ALGEBRA LESSON 4 -2 Suppose a certain star is 104 light-years from Earth. How many light-years is that? 104 = 10 • 10 = 10, 000 light-years The exponent indicates that the base 10 is used as a factor 4 times. Multiply. Quick Check 4 -2
Exponents PRE-ALGEBRA LESSON 4 -2 a. Simplify 3(1 + 4)3 = 3(5)3 Work within parentheses first. = 3 • 125 Simplify 53. = 375 Multiply. b. Evaluate 7(w + 3)3 + z, for w = – 5 and z = 6. 7(w + 3)3 + z = 7(– 5 + 3)3 + 6 Replace w with – 5 and z with 6. = 7(– 2)3 + 6 Work within parentheses. = 7(– 8) + 6 Simplify (– 2)3. = – 56 + 6 Multiply from left to right. = – 50 Add. 4 -2 Quick Check
Exponents PRE-ALGEBRA LESSON 4 -2 Write using exponents. 1. x • y • z • x • z 2. a • b • b • 3 x 2 yz 2 3 ab 3 3. Simplify 5(2 + 4)2. 180 4. Evaluate (g 3 – 7)2 • 5 + 4, for g = 3. 2, 004 4 -2
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 Find three integers whose sum is 12 and whose product is 42. 2, 3, 7 4 -3
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 (For help, go to Lesson 4 -1. ) List the positive factors of each number. 1. 15 2. 35 3. 7 4. 20 5. 100 6. 121 Check Skills You’ll Need 4 -3
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 Solutions 1. 1 • 15, 3 • 5; 1, 3, 5, 15 2. 3. 4. 5. 6. 1 • 35, 5 • 7; 1, 5, 7, 35 1 • 7; 1, 7 1 • 20, 2 • 10, 4 • 5; 1, 2, 4, 5, 10, 20 1 • 100, 2 • 50, 4 • 25, 5 • 20, 10 • 10; 1, 2, 4, 5, 10, 25, 50, 100 1 • 121, 11 • 11; 1, 121 4 -3
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 State whether each number is prime or composite. Explain. a. 46 Composite; 46 has more than two factors, 1, 2, 23, and 46. b. 13 Prime; 13 has exactly 2 factors, 1 and 13. Quick Check 4 -3
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 Use a factor tree to write the prime factorization of 273 Prime 3 • 91 7 • 13 Start with a prime factor. Continue branching. Stop when all factors are prime. 3 • 7 • 13 Write the prime factorization. 273 = 3 • 7 • 13 Quick Check 4 -3
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 Find the GCF of each pair of numbers or expressions. a. 24 and 30 24 = 23 • 3 Write the prime factorizations. 30 = 2 • 3 • 5 Find the common factors. GCF = 2 • 3 Use the lesser power of the common factors. = 6 The GCF of 24 and 30 is 6. b. 36 ab 2 and 81 b 36 ab 2 = 22 • 32 • a • b 2 Write the prime factorizations. 81 b = 34 • b Find the common factors. GCF = 32 • b Use the lesser power of the common factors. = 9 b The GCF of 36 ab 2 and 81 b is 9 b. 4 -3 Quick Check
Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4 -3 Tell whether each number is prime or composite. 1. 123 2. 47 composite prime Write the prime factorization for each number. 3. 64 4. 45 26 32 • 5 Find the GCF for each pair. 5. 80 and 120 40 6. 62 b 3 c 2 d and 31 b 2 c 3 d 31 b 2 c 2 d 4 -3
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 Find the GCF for each pair of numbers. a. 12 and 18 b. 15 and 60 c. 54 and 60 6 15 6 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 (For help, go to Lesson 4 -3. ) Find each GCF. 1. 14, 21 3. 5 mn, 15 m 2 n 2. 48, 60 4. 63 r 2, 48 s 3 Check Skills You’ll Need 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 Solutions 1. 14, 21 14 = 2 • 7 21 = 3 • 7 GCF = 7 2. 48, 60 48 = 24 • 3 60 = 22 • 3 • 5 GCF = 22 • 3 = 12 3. 5 mn, 15 m 2 n 5 mn = 5 • m • n 15 m 2 n = 3 • 5 • m 2 • n GCF = 5 • m • n = 5 mn 4. 63 r 2, 48 s 3 63 r 2 = 32 • 7 • r 2 48 s 3 = 24 • 3 • s 3 GCF = 3 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 18 Find two fractions equivalent to . 21 a. 18 21 18 • 2 = 21 • 2 36 = 42 b. 18 21 = 18 ÷ 3 21 ÷ 3 = 6 7 36 42 18 21 The fractions and are both equivalent to . Quick Check 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 You learn that 21 out of the 28 students in a class, 21 or , buy their lunches in the cafeteria. Write this fraction in 28 simplest form. The GCF of 21 and 28 is 7. 21 = 21 ÷ 7 28 28 ÷ 7 = 3 4 Divide the numerator and denominator by the GCF, 7. Simplify. 3 of the students in the class buy their lunches in the cafeteria. 4 Quick Check 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 Write in simplest form. p a. 2 p p p 1 = 1 2 p 2 p = 1 2 Divide the numerator and denominator by the common factor, p. Simplify. 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 (continued) b. 14 q 2 rs 3 8 qrs 2 = 2 • 7 • q • r • s • s 2 • 2 • q • r • s Write as a product of prime factors. = 21 • 7 • q 1 • q • r 1 • s 21 • 2 • q 1 • r 1 • s 1 Divide the numerator and denominator by the common factors. = 7 • q • s 2 • 2 Simplify. = 7 • q • s 4 Simplify. = 7 qs 4 Quick Check 4 -4
Simplifying Fractions PRE-ALGEBRA LESSON 4 -4 Find two fractions equivalent to each fraction. 11 7 1. 2. 16 21 Sample answer: 22 33 and 48 32 1 10 and 30 3 Write in simplest form. 3. 13 52 1 4 4 8 4. wx y 2 3 wxy xy 7 w 4 -4
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 Choose the symbol <, = , or > that makes each statement true. 3 5 7 a. 2 + 1 + 2 ? 4 + 1 1 8 8 5 1 b. 3 + 2 + 9 ? 4 + 3 3 6 = < 4 -5 10 10 4
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 (For help, go to Skills Handbook p. 775. ) Compare. Use > to < to complete each statement. 1. 3 0 2. – 16 – 25 3. 0 1 4. – 30 – 20 Check Skills You’ll Need 4 -5
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 (For help, go to Skills Handbook p. 775. ) Solutions 1. 3 > 0 3. 0 < 1 2. – 16 > – 25 4. – 30 < – 20 4 -5
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 Aaron, Chris, Maria, Sonia, and Ling are on a class committee. They want to choose two members to present their conclusions to the class. How many different groups of two members can they form? 4 -5
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 (continued) First, pair Aaron with each of the four other committee members. Next, pair Chris with each of the three members left. Since Aaron and Chris have already been paired, you don’t need to count them again. Repeat for the rest of the committee members. Each successive tree has one less branch. Aaron Chris Maria Sonia Ling There are 10 different groups of two committee members. 4 -5 Quick Check
Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4 -5 Solve each problem. 1. Twelve people are at a party. Each person greets each of the other persons exactly once. How many greetings will there be in all? 66 2. How many different pairs of classmates can you choose from six classmates? 15 pairs 3. Each small box is a square. What is the number of different squares shown? 17 4 -5
Rational Numbers PRE-ALGEBRA LESSON 4 -6 29, 716 Write in simplest form. 52, 003 4 7 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 (For help, go to Lesson 4 -4. ) Write in simplest form. 1. 2 10 3. 28 35 2. 14 21 4. 6 8 Check Skills You’ll Need 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 Solutions 2 2 ÷ 2 1 1. = 10 ÷ 2 = 5 10 3. 28 = 28 ÷ 7 = 4 35 35 ÷ 7 5 2. 14 14 ÷ 7 2 = 21 ÷ 7 = 3 21 4. 6 ÷ 2 6 3 = 8 ÷ 2 = 4 8 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 2 Write two lists of fractions equivalent to . 3 2 4 6 = = = … Numerators and denominators are positive. 3 6 9 2 – 4 = = … Numerators and denominators are negative. 3 – 6 Quick Check 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 Graph each rational number on a number line. 3 a. – 4 b. 0. 5 c. 0 d. 1 3 Quick Check 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 A fast sports car can accelerate from a stop to 90 ft/s in 5 seconds. What is its acceleration in feet per second per f–i second (ft/s 2)? Use the formula a = , where a is t acceleration, f is final speed, i is initial speed, and t is time. f–i a = t = 90 – 0 5 90 Use the acceleration formula. Substitute. = 5 Subtract. = 18 Write in simplest form. The car’s acceleration is 18 ft/s 2. Quick Check 4 -6
Rational Numbers PRE-ALGEBRA LESSON 4 -6 Write three fractions equivalent to the given fraction. 5 – 5 10 15 1. 6 Sample: , , – 6 12 18 Graph each rational number on a number line. 2. a. 1 b. – 1 c. 1. 5 4 2 d. 0. 4 3. A car can accelerate from 0 to 70 ft/s in 5 s. What is the acceleration of the car in feet per second (ft/s 2)? 14 ft/s 2 4 -6
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 23 32 is the prime factorization for __. ? 72 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 (For help, go to Lesson 4 -2. ) Write using exponents. 1. k • k • k 3. 2 • 2 • 2 2. m • n • m • n 4. 5 • 5 Check Skills You’ll Need 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 Solutions 1. k • k • k = k 4 3. 2 • 2 • 2 = 24 2. m • n • m • n = m • n • n = m 2 n 2 4. 5 • 5 = 53 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 Simplify each expression. a. 52 • 53 = 52 + 3 Add the exponents of powers with the same base. = 55 = 3, 125 Simplify. b. x 5 • x 7 • y 2 • y = x 5 + 7 • y 2 + 1 Add the exponents of powers with the same base. = x 12 y 3 Simplify. Quick Check 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 Simplify 3 a 3 • (– 5 a 4) = 3 • (– 5) • a 3 • a 4 Use the Commutative Property of Multiplication. = – 15 a 3 + 4 Add the exponents. = – 15 a 7 Simplify. Quick Check 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 Simplify each expression. a. (23)3 = (2)3 • 3 Multiply the exponents. = (2)9 Simplify the exponent. = 512 Simplify. b. (g 5)4 = g 5 • 4 = g 20 Multiply the exponents. Simplify the exponent. Quick Check 4 -7
Exponents and Multiplication PRE-ALGEBRA LESSON 4 -7 Simplify each expression. 1. 22 • 23 2. g 2 • h 4 • h 32 3. 4 r 6 s • 7 r 3 s 5 28 r 9 s 6 5. (v 3)8 g 2 • h 7 4. –(22)5 – 1, 024 v 24 4 -7
Exponents and Division PRE-ALGEBRA LESSON 4 -8 Mari can package 14 seashells in a box. She has 360 seashells. How many full boxes does she have? How many shells are left over after the boxes are filled? 25; 10 4 -8
Exponents and Division PRE-ALGEBRA LESSON 4 -8 (For help, go to Lesson 4 -4. ) Write in simplest form. x 2 1. x 6 xy 3. 9 y 2. y y 2 4. 4 ab 2 16 b Check Skills You’ll Need 4 -8
Exponents and Division PRE-ALGEBRA LESSON 4 -8 (For help, go to Lesson 4 -4. ) Solutions 1 x • x x 2 1. x = x 1 2. y 2 = y y y • y 1 1 = y 1 1 3. 6 xy = 2 • 3 • x • y = 2 x 3 • y 9 y 3 1 1 1 2 ab 4. 4 ab = 2 • a • b 1 = a • b = 4 16 b 2 • 2 • b 2 • 2 1 1 4 -8
Exponents and Division PRE-ALGEBRA LESSON 4 -8 Simplify each expression. a. 412 48 Subtract the exponents. = 44 Simplify the exponent. = 256 b. = 412 – 8 Simplify. w 18 w 13 w 18 = w 18 – 13 w 13 = w 5 Subtract the exponents. Simplify the exponent. Quick Check 4 -8
Exponents and Division PRE-ALGEBRA LESSON 4 -8 Simplify each expression. 73 a. (– 12)73 73 – 73 73 = (– 12)0 = 1 b. 8 s 20 32 s 20 1 = 4 s 0 1 = 4 • 1 1 = 4 Subtract the exponents. Simplify. 8 32 Subtract the exponents. Simplify s 0. Multiply. 4 -8 Quick Check
Exponents and Division PRE-ALGEBRA LESSON 4 -8 Simplify each expression. 12 a. 614 6 612 = 612 – 14 614 Subtract the exponents. = 6– 2 1 62 1 = 36 = Write with a positive exponent. Simplify. 4 b. z 15 z z 4 = z 4 – 15 z 15 Subtract the exponents. = z– 11 = 1 11 z Write with a positive exponent. 4 -8 Quick Check
Exponents and Division PRE-ALGEBRA LESSON 4 -8 a 2 b 3 Write without a fraction bar. ab 15 a 2 b 3 2 – 1 b 3 – 15 Use the rule for Dividing Powers with the 15 = a ab Same Base. = ab– 12 Subtract the exponents. Quick Check 4 -8
Exponents and Division PRE-ALGEBRA LESSON 4 -8 Simplify each expression. – 15 b 5 c 3 75 1. 3 2. 60 b 3 c 2 7 1 49 – b 2 c 4 6 n 3. Write without a fraction bar. 24 14 3 n 2 n– 10 4 -8
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Estimate in years the age of someone who is one million minutes old. about 2 years 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 (For help, go to Lesson 4 -7. ) Write each expression with a simple exponent. 1. 103 • 105 2. 107 • 109 3. 105 • 10– 3 4. 10– 6 • 103 Check Skills You’ll Need 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 (For help, go to Lesson 4 -7. ) Solutions 1. 103 • 105 = 103+5 2. 107 • 109 = 107+9 = 108 3. 105 • 10– 3 = 105– 3 = 1016 4. 10– 6 • 103 = 10– 6+3 = 102 = 10– 3 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 About 6, 300, 000 people visited the Eiffel Tower in the year 2000. Write this number in scientific notation. 6, 300, 000 6 places 6. 3 106 Move the decimal point to get a decimal greater than 1 but less than 10. Drop the zeros after the 3. You moved the decimal point 6 places. The number is large. Use 6 as the exponent of 10. Quick Check 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Write 0. 00037 in scientific notation. 0. 00037 4 places 3. 7 10– 4 Move the decimal point to get a decimal greater than 1 but less than 10. Drop the zeros before the 3. You moved the decimal point 4 places. The number is small. Use – 4 as the exponent of 10. Quick Check 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Write each number in standard notation. a. 3. 6 104 3. 6000 36, 000 b. 7. 2 10– 3 007. 2 0. 0072 Write zeros while moving the decimal point. Rewrite in standard notation. Quick Check 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Write each number in scientific notation. a. 0. 107 1012 = 1. 07 10– 1 1012 = 1. 07 1011 Write 0. 107 as 1. 07 10– 1. Add the exponents. b. 515. 2 10– 4 = 5. 152 10– 2 Write 515. 2 as 5. 152 102. Add the exponents. Quick Check 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Order 0. 035 104, 710 10– 1, and 0. 69 102 from least to greatest. Write each number in scientific notation. 0. 035 104 3. 5 102 710 10– 1 0. 69 102 7. 1 10 6. 9 10 Order the powers of 10. Arrange the decimals with the same power of 10 in order. 6. 9 10 7. 1 10 3. 5 102 Write the original numbers in order. 0. 69 102, 710 10– 1, 0. 035 104 4 -9 Quick Check
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Multiply 4 10– 6 and 7 109. Express the result in scientific notation. (4 10– 6)(7 109) = 4 7 10– 6 109 Use the Commutative Property of Multiplication. = 28 10– 6 109 Multiply 4 and 7. = 28 103 Add the exponents. = 2. 8 101 103 Write 28 as 2. 8 101. = 2. 8 104 Add the exponents. Quick Check 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Quick Check In chemistry, one mole of any element contains approximately 6. 02 1023 atoms. If each hydrogen atom weighs approximately 1. 67 10– 27 kg, approximately how much does one mole of hydrogen atoms weigh? Multiply number of atoms by (6. 02 1023)(1. 67 10– 27) weight of each. = 6. 02 1. 67 1023 10– 27 10. 1 1023 10– 27 Use the Commutative Property of Multiplication. Multiply 6. 02 and 1. 67. = 10. 1 10– 4 Add the exponents. = 1. 01 10– 4 Write 10. 1 as 1. 01 101. = 1. 01 10– 3 Add the exponents. One mole of hydrogen atoms weighs approximately 1. 01 10– 3 kg. 4 -9
Scientific Notation PRE-ALGEBRA LESSON 4 -9 Write each number in scientific notation. 1. 5, 400, 000 2. 0. 0000867 5. 4 106 8. 67 10– 5 Write each number in standard notation. 3. 3. 45 106 3, 450, 000 4. 1. 99 10– 5 0. 0000199 5. Order 7. 2 105, 7. 2 106, 7. 02 106, and 7. 1 10– 6 from least to greatest. 7. 1 10– 6, 7. 2 105, 7. 02 106, 7. 2 106 6. Multiply 14 106 and 4 10– 4. Express the result in scientific notation. 5. 6 103 4 -9
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